= Solution
The element $-1_R$ lies in the <unit group> and satisfies $(-1_R)^2=1_R$. Its <order of a group element> divides two and, by <Lagrange's theorem>, divides the odd number $|U(R)|$. Its order is therefore one, so $-1_R=1_R$ and $2\cdot1_R=0$.
It follows that
$$
\boxed{\mathbb F_2\longrightarrow R,\qquad\overline a\longmapsto a\cdot1_R}
$$
is a well-defined unital <ring homomorphism>. For a nonzero <ring>, this says its <characteristic of a ring> is two. If the <zero ring> is permitted, the same map still exists. This is the <odd unit group forces characteristic two> argument, with that harmless exception to the characteristic wording.
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